Problem: For $-1<r<1$, let $S(r)$ denote the sum of the geometric series \[12+12r+12r^2+12r^3+\cdots .\]Let $a$ between $-1$ and $1$ satisfy $S(a)S(-a)=2016$. Find $S(a)+S(-a)$.
Answer: By the formula for an infinite geometric series, \[S(r) = \frac{12}{1-r}.\]Thus, we are given that \[S(a)S(-a) = \frac{12}{1-a} \cdot \frac{12}{1+a} = \frac{144}{1-a^2} = 2016.\]Instead of solving for $a$ explicitly, we note that \[\begin{aligned} S(a) + S(-a) &= \frac{12}{1-a} + \frac{12}{1+a}\\& = \frac{12(1-a)+12(1+a)}{1-a^2}\\& = \frac{24}{1-a^2}\\& = \frac{1}{6} \cdot \frac{144}{1-a^2} \\&= \frac{1}{6} \cdot 2016\\& = \boxed{336},\end{aligned}\]which is the answer.